There remain obstacles to be handled to obtain a \(2^{O(k^2)}n\)  time algorithm.
First, due to properties of the backbone structure we defined, we can show that
the length of \(\bar{f}(F)\)  of all initial functions \(f\)  constructed from [1]} is \(2^{O(k)}\) .
But even in our case, \(|\Pi |\)  can be \(2^{\Theta (k)}n\)  as the total number of edges of the walks of \(\mathcal {W}\)  can be \(2^{\Theta (k)}n\) .
We resolve this issue by designing a suitable data structure and implementing the algorithm of [1]}
using this data structure.
In this way, we can fully overcome the third obstacle: Given a correct weak \(T\) -linkage of \(G\)  defined with respect to the backbone structure (frames and skeleton forests),
we can compute a \(T\) -linkage of \(G\)  in \(2^{O(k)}n\)  time.
